Why are descriptive statistics important?
In January 1986, the space shuttle Challenger broke apart shortly after liftoff. The
accident was caused by a part that was not designed to fly at the unusually cold
temperature of 29◦ F at launch.
6
4
2
0
Frequency
8
10
Here are the launch-temperatures of the first 25 shuttle missions (in degrees F):
66,70,69,80,68,67,72,70,70,57,63,70,78,67,53,67,75,70,81,76,79,75,76,58,29
20
30
40
50
60
Temperature
70
80
90
The two most important functions of descriptive statistics are:
I
Communicate information
I
Support reasoning about data
When exploring data of large size, it becomes essential to use summaries.
Graphical summaries of data
It is best to use a graphical summary to communicate information, because people
prefer to look at pictures rather than at numbers.
There are many ways to visualize data. The nature of the data and the goal of the
visualization determine which method to choose.
Pie chart and dot plot
California
International
Other US
Washington
Oregon
International
Oregon
Washington
Other US
California
0
10
20
30
40
Percent
The dot plot makes it easier to compare frequencies of various categories, while the pie
chart allows more easily to eyeball what fraction of the total a category corresponds to.
Bar graph
When the data are quantitative (i.e. numbers), then they should be put on a number
line. This is because the ordering and the distance between the numbers convey
important information.
0
2
4
6
8
The bar graph is essentially a dot plot put on its side.
3
4
5
Number of assignments completed
6
The histogram
The histogram allows to use blocks with different widths.
Key point: The areas of the blocks are
proportional to frequency.
So the percentage falling into a block can be figured without a vertical scale since the
total area equals 100%.
But it’s helpful to have a vertical scale (density scale). Its unit is ‘% per unit’, so in the
above example the vertical unit is ‘% per year’.
The histogram gives two kinds of information about the data:
1. Density (crowding): The height of the bar tells how many subjects there are for one
unit on the horizontal scale. For example, the highest density is around age 19 as
.04 = 4% of all subjects are age 19. In contrast, only about 0.7% of subjects fall into
each one year range for ages 60–80.
2. Percentages (relative frequences): Those are given by
area = height x width.
For example, about 14% of all subjects fall into the age range 60–80, because the
corresponding area is (20 years) x (0.7 % per year)=14 %. Alternatively, you can find
this answer by eyeballing that this area makes up roughly 1/7 of the total area of the
histogram, so roughly 1/7=14% of all subjects fall in that range.
The boxplot (box-and-whisker plot)
25
20
15
10
Miles per gallon for 32 cars
30
The boxplot depicts five key numbers of the data:
25
20
15
10
Miles per gallon
30
The boxplot conveys less information than a histogram, but it takes up less space and
so is well suited to compare several datasets:
4
6
Number of Cylinders
8
The scatterplot
The scatterplot is used to depict data that come as pairs.
25000
20000
15000
Income
The scatterplot visualizes the relationship
between the two variables.
10000
5000
0
6
8
10
12
Education
14
16
Providing context is important
Statistical analyses typically compare the observed data to a reference. Therefore
context is essential for graphical integrity.
I
‘The Visual Display of Quantitative Information’ by Edward Tufte (p.74)
One way to provide context is by using small multiples. The compact design of the
boxplot makes it well suited for this task:
Providing context with small multiples
ale up the projects so it can
elf," Webb said.
gether experts from manyse cooperation results in a
over the 2000 figure.
"The total is surprisinglylarge in
light of the overall sharp decline in
stock market values over this periogical breakthroughs.
od," RAND's Council for Aid to Edis heading towards interucation said. The non-profit CounWebb said. "Having the
Sophisticated software
makes
it tempting
cil forAid
has trackedto
to Education
biologists all in the same
roughs. I think it is a good
erdisciplinary program
at
of cooperation between dising funding requirements.
fficult because it is generalexperiments. Often bioloogy development is critical,
ding. You are going after
al sources that are used to
al way," Webb said.
fense Advanced Research
n more willing to take the
ayoff projects, according to
rtment of Defense funding
ects that are long-range,
ard, like those taken on by
I
"1 expected growth in giving
by foundations to be there, but 1
didn't expect it to be nearly as
high as it was," Kaplan said.
"There was so much bad news in
the stock market, especially durproduce
showy
but
poor
period,
with
the
ing that fiscal
Pitfalls when visualizing data
dent Hennessy in 2000 with th
goal to raise $ 1 billion over five
years.
visualizations:
This is "the largest campaign
specifically for undergraduat
education ever undertaken b
any university," Henness
wrote in a 2000 letter introducing the program.
As of March 31, Stanford
campaign had already raise
$733 million, although a portio
of that total reflected as-yet unme
commitments to match dona
tions.
Columbia University, wi
$359 million in 2001, was just be
hind Stanford. Indiana Universi
ty was the most-funded pub
university, in seventh place wi
$301 million.
AARON STAPLE/The Stanford Daily
The ‘Ghettysburg Powerpoint Presentation’ by Peter Norvig
Steakburger"
ority now is its Campaign fo
Undergraduate Education
launched by University Presi-
Numerical summary measures
For summerizing data with one number, use the mean (=average) or the median.
The median is the number that is larger than half the data and smaller than the other
half.
Mean vs. median
Mean and median are the same when the histogram is symmetric.
0
5
10
15
20
25
30
100 measurements of the speed of light
600
700
800
900
km/sec
1000
1100
Mean vs. median
When the histogram is skewed to the right, then the mean can be much larger than the
median.
So if the histogram is very skewed, then use the median.
Mean vs. median
If the median sales price of 10 homes is $ 1 million, then we know that 5 homes sold for
$ 1 million or more.
If we are told that the average sale price is $ 1 million, then we can’t draw such a
conclusion:
Percentiles
The 90th percentile of incomes is
$ 135,000: 90% of households report an
income of $ 135,000 or less, 10% report
more.
The 75th percentile is called 3rd quartile: $ 85,000
The 50th percentile is the median: $ 50,000
The 25th percentile is called 1st quartile.
Five-number summary
25
20
10
15
Miles per gallon for 32 cars
30
Recall that the boxplot gives a five-number summary of the data:
the smallest number, 1st quartile, median, 3rd quartile, largest number.
The interquartile range = 3rd quartile − 1st quartile.
It measures how spread out the data are.
The standard deviation
A more commonly used measure of spread is the standard deviation.
x̄ stands for the average of the numbers x1 , . . . , xn .
The standard deviation of these numbers is
v
u n
u1 X
s = t
(xi − x̄)2 or
n
i=1
v
u
u
t
n
1 X
(xi − x̄)2
n−1
i=1
The two numbers x̄ and s are often used to summarize data. Both are sensitive to a
few large or small data.
If that is a concern, use the median and the interquartile range.